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Title:On the distribution of time-to-proof of mathematical conjectures

Abstract: What is the productivity of Science? Can we measure an evolution of the
production of mathematicians over history? Can we predict the waiting time till
the proof of a challenging conjecture such as the P-versus-NP problem?
Motivated by these questions, we revisit a suggestion published recently and
debated in the "New Scientist" that the historical distribution of
time-to-proof's, i.e., of waiting times between formulation of a mathematical
conjecture and its proof, can be quantified and gives meaningful insights in
the future development of still open conjectures. We find however evidence that
the mathematical process of creation is too much non-stationary, with too
little data and constraints, to allow for a meaningful conclusion. In
particular, the approximate unsteady exponential growth of human population,
and arguably that of mathematicians, essentially hides the true distribution.
Another issue is the incompleteness of the dataset available. In conclusion we
cannot really reject the simplest model of an exponential rate of conjecture
proof with a rate of 0.01/year for the dataset that we have studied,
translating into an average waiting time to proof of 100 years. We hope that
the presented methodology, combining the mathematics of recurrent processes,
linking proved and still open conjectures, with different empirical
constraints, will be useful for other similar investigations probing the
productivity associated with mankind growth and creativity.